60.9.38 problem 1893
Internal
problem
ID
[11892]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1893
Date
solved
:
Tuesday, January 28, 2025 at 06:24:02 PM
CAS
classification
:
system_of_ODEs
\begin{align*} a_{1} \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+b_{1} \left (\frac {d}{d t}x \left (t \right )\right )+c_{1} x \left (t \right )-A \left (\frac {d}{d t}y \left (t \right )\right )&=B \,{\mathrm e}^{i \omega t}\\ a_{2} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+b_{2} \left (\frac {d}{d t}y \left (t \right )\right )+c_{2} y \left (t \right )+A \left (\frac {d}{d t}x \left (t \right )\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.919 (sec). Leaf size: 1558
dsolve([a__1*diff(x(t),t,t)+b__1*diff(x(t),t)+c__1*x(t)-A*diff(y(t),t)=B*exp(I*omega*t),a__2*diff(y(t),t,t)+b__2*diff(y(t),t)+c__2*y(t)+A*diff(x(t),t)=0],singsol=all)
\begin{align*}
\text {Expression too large to display} \\
y \left (t \right ) &= \frac {i {\mathrm e}^{i \omega t} \omega A B}{-a_{1} a_{2} \omega ^{4}+i a_{1} b_{2} \omega ^{3}+i a_{2} b_{1} \omega ^{3}+A^{2} \omega ^{2}+a_{1} c_{2} \omega ^{2}+a_{2} c_{1} \omega ^{2}+b_{1} b_{2} \omega ^{2}-b_{1} c_{2} \omega i-b_{2} c_{1} \omega i-c_{2} c_{1}}+c_3 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +c_{1} a_{2} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} =1\right ) t}+c_4 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +c_{1} a_{2} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} =2\right ) t}+c_5 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +c_{1} a_{2} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} =3\right ) t}+c_6 \,{\mathrm e}^{\operatorname {RootOf}\left (a_{1} a_{2} \textit {\_Z}^{4}+\left (a_{1} b_{2} +a_{2} b_{1} \right ) \textit {\_Z}^{3}+\left (A^{2}+c_{2} a_{1} +c_{1} a_{2} +b_{1} b_{2} \right ) \textit {\_Z}^{2}+\left (b_{1} c_{2} +b_{2} c_{1} \right ) \textit {\_Z} +c_{2} c_{1} , \operatorname {index} =4\right ) t} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.204 (sec). Leaf size: 5546
DSolve[{a1*D[x[t],{t,2}]+b1*D[x[t],t]+c1*x[t]-A*D[y[t],t]==B*Exp[I*\[Omega]*t],a2*D[y[t],{t,2}]+b2*D[y[t],t]+c2*y[t]+A*D[x[t],t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
Too large to display